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Unformatted text preview: Gradient operator • In our calculation of d φ along the vector ~ ds , we see that it can be described as the scalar product d φ = ∂φ ∂ x ˆ i + ∂φ ∂ y ˆ j + ∂φ ∂ z ˆ k · u x ds ˆ i + u y ds ˆ j + u z ds ˆ k • We take d φ = ∇ φ · ~ ds = ds ∇ φ · ~ u and hence define the gradient operator (in a Cartesian system) ~ ∇ φ = grad φ = ∂φ ∂ x ˆ i + ∂φ ∂ y ˆ j + ∂φ ∂ z ˆ k • We then define the directional derivative as d φ ds = ~ ∇ φ · ~ u Patrick K. Schelling Introduction to Theoretical Methods Example of gradient operators and directional derivatives • As an example, consider Section 6, problems 11 and 12 • When φ ( x , y , z ) is the electrostatic potential, the electric field ~ E ( x , y , z ) =∇ φ • For φ ( x , y ) = x 2 y 2 , compute the field ~ E ( x , y ) (There is no dependence on z ) ~ E ( x , y ) =∇ φ ( x , y ) = ∂φ ∂ x ˆ i ∂φ ∂ y ˆ j = 2 x ˆ i + 2 y ˆ j • Sketch the equipotential lines, and the direction and magnitude of the vector field at various points Patrick K. Schelling Introduction to Theoretical Methods Example, continued • Find the rate of change if φ with distance ds at (1 , 2) along direction 3 ˆ i ˆ j • The direction as a unit vector is ~ u = 3 √ 10 ˆ i 1 √ 10 ˆ j • At x = 1, y = 2, ∇ φ = 2 ˆ i 4 ˆ j , so d φ ds = 2 ˆ i 4 ˆ j · 3 √ 10 ˆ i 1 √ 10 ˆ j = 2 √ 10 Patrick K. Schelling Introduction to Theoretical Methods Gradient operator in polar coordinates • We can determine the gradient operator in a polar coordinate system, so φ ( r ,θ ) • In polar coordinates, ~ ds = dr ˆ e r + rd θ ˆ e θ • We have that d φ = ∂φ ∂ r dr + ∂φ ∂θ d θ = ∇ φ · ~ ds • We take for ∇ φ , ∇ φ = ∂φ ∂ r ˆ e r + 1 r ∂φ ∂θ ˆ e θ • Then we can see d φ = ∇ φ · ~ ds = ∂φ ∂ r ˆ e r + 1 r ∂φ ∂θ ˆ e θ · ( dr ˆ e r + rd θ ˆ e θ ) = ∂φ ∂ r dr + ∂φ ∂θ d θ Patrick K. Schelling Introduction to Theoretical Methods Example: Section 6, Problem 17 • Find ∇ r with r = p x 2 + y 2 , and write the answer in polar and Cartesian coordinates and verify that they are the same • We can take φ ( r ,θ ) = r and use ∇ φ = ∂φ ∂ r ˆ e r + 1 r ∂φ ∂θ ˆ e θ , ∇ r = ˆ e r • Now in Cartesian coordinates, take φ ( x , y ) = p x 2 + y 2 and use ∇ φ ( x , y ) = ∂φ ∂ x ˆ i + ∂φ ∂ y ˆ j ∇ p x 2 + y 2 = x ˆ i + y ˆ j p x 2 + y 2 Patrick K. Schelling Introduction to Theoretical Methods Example continued • Now we can prove that ˆ e r = x ˆ i + y...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.
 Spring '03
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